Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

Fuzzy implication functions based on powers of continuous t-norms

The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions.Peer ReviewedPostprint (author's final draft

The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ⊛ on the set II of fuzzy implications they obtained, for the first time, a monoid structure (I,⊛)(I,⊛) on the set II. Some algebraic aspects of (I,⊛)(I,⊛) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51–67). However, the properties of fuzzy implications generated or obtained using the ⊛-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles , the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. ⊛ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ⊛, specifically the families of (S,N)(S,N)-, R-, f- and g-implications, are studied. This study shows that the ⊛-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ⊛-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ⊛-composition

Fuzzy implications: alpha migrativity and generalised laws of importation

In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I(C(x,α),y)=I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(x,α),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related

Some characterizations of T-power based implications

Recently, the so-called family of T-power based implications was introduced. These operators involve the use of Zadeh’s quantifiers based on powers of t-norms in its definition. Due to the fact that Zadeh’s quantifiers constitute the usual method to modify fuzzy propositions, this family of fuzzy implication functions satisfies an important property in approximate reasoning such as the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, an in-depth analysis of this property is performed by characterizing all binary functions satisfying it. From this general result, a fully characterization of the family of T-power based implications is presented. Furthermore, a second characterization is also proved in which surprisingly the invariance property is not explicitly used.Peer ReviewedPostprint (author's final draft

Homomorphisms on the monoid of fuzzy implications and the iterative functional equation I(x,I(x,y))=I(x,y)

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛⊛-composition, from a given pair of fuzzy implications [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67]. However, as with any generation process, the ⊛⊛-composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛⊛-composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see [Information Sciences 177, 2954–2970 (2007); 180, 2487–2497 (2010); 186, 209–221 (2012)]. In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛⊛-composition as a binary operation ⊛⊛ on the set II of all fuzzy implications one obtains a monoid structure (I,⊛)(I,⊛) on the set II. From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set KK of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid (I,⊛I,⊛) and show that KK not only answers our questions regarding the generative power of the ⊛⊛-composition but also contains many as yet unknown solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y)

The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity

Bandler-Kohout Subproduct with Yager’s Families of Fuzzy Implications: A Comprehensive Study

Approximate reasoning schemes involving fuzzy sets are one of the best known applications of fuzzy logic in the wider sense. Fuzzy Inference Systems (FIS) or Fuzzy Inference Mechanisms (FIM) have many degrees of freedom, viz., the underlying fuzzy partition of the input and output spaces, the fuzzy logic operations employed, the fuzzification and defuzzification mechanism used, etc. This freedom gives rise to a variety of FIS with differing capabilities

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book